scholarly journals FIRST-ORDER AXIOMATISATIONS OF REPRESENTABLE RELATION ALGEBRAS NEED FORMULAS OF UNBOUNDED QUANTIFIER DEPTH

2021 ◽  
pp. 1-18
Author(s):  
ROB EGROT ◽  
ROBIN HIRSCH
Keyword(s):  
Relation Algebras ◽  
First Order ◽  
Representable Relation

Nonfinite axiomatizability results for cylindric and relation algebras

1989 ◽  
Vol 54 (3) ◽  
pp. 951-974 ◽  
Author(s):  
Roger D. Maddux
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Algebraic Semantics ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
First Order ◽  
Finite Set ◽  
Representable Relation

AbstractThe set of equations which use only one variable and hold in all representable relation algebras cannot be derived from any finite set of equations true in all representable relation algebras. Similar results hold for cylindric algebras and for logic with finitely many variables. The main tools are a construction of nonrepresentable one-generated relation algebras, a method for obtaining cylindric algebras from relation algebras, and the use of relation algebras in defining algebraic semantics for first-order logic.

scholarly journals THE VARIETY OF COSET RELATION ALGEBRAS

2018 ◽  
Vol 83 (04) ◽  
pp. 1595-1609 ◽  
Author(s):  
STEVEN GIVANT ◽  
HAJNAL ANDRÉKA
Keyword(s):  
Large Class ◽  
Identity Element ◽  
Relation Algebra ◽  
Order Theory ◽  
Relation Algebras ◽  
First Order ◽  
First Order Theory ◽  
Atomic Pair ◽  
Finite Set ◽  
Image Position

AbstractGivant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Non-finite-axiomatizability results in algebraic logic

1992 ◽  
Vol 57 (3) ◽  
pp. 832-843 ◽  
Author(s):  
Balázs Biró
Keyword(s):  
Algebraic Logic ◽  
Relation Algebra ◽  
Equational Theory ◽  
Negative Answer ◽  
Relation Algebras ◽  
Representable Relation Algebra ◽  
Finite Set ◽  
Variable Symbol ◽  
Axiom Systems ◽  
Representable Relation

This paper deals with relation, cylindric and polyadic equality algebras. First of all it addresses a problem of B. Jónsson. He asked whether relation set algebras can be expanded by finitely many new operations in a “reasonable” way so that the class of these expansions would possess a finite equational base. The present paper gives a negative answer to this problem: Our main theorem states that whenever Rs+ is a class that consists of expansions of relation set algebras such that each operation of Rs+ is logical in Jónsson's sense, i.e., is the algebraic counterpart of some (derived) connective of first-order logic, then the equational theory of Rs+ has no finite axiom systems. Similar results are stated for the other classes mentioned above. As a corollary to this theorem we can solve a problem of Tarski and Givant [87], Namely, we claim that the valid formulas of certain languages cannot be axiomatized by a finite set of logical axiom schemes. At the same time we give a negative solution for a version of a problem of Henkin and Monk [74] (cf. also Monk [70] and Németi [89]).Throughout we use the terminology, notation and results of Henkin, Monk, Tarski [71] and [85]. We also use results of Maddux [89a].Notation. RA denotes the class of relation algebras, Rs denotes the class of relation set algebras and RRA is the class of representable relation algebras, i.e. the class of subdirect products of relation set algebras. The symbols RA, Rs and RRA abbreviate also the expressions relation algebra, relation set algebra and representable relation algebra, respectively.For any class C of similar algebras EqC is the set of identities that hold in C, while Eq1C is the set of those identities in EqC that contain at most one variable symbol. (We note that Henkin et al. [85] uses the symbol EqC in another sense.)

Relation algebra reducts of cylindric algebras and an application to proof theory

2002 ◽  
Vol 67 (1) ◽  
pp. 197-213 ◽  
Author(s):  
Robin Hirsch ◽  
Ian Hodkinson ◽  
Roger D. Maddux
Keyword(s):  
Proof Theory ◽  
Predicate Logic ◽  
Algebraic Logic ◽  
Relation Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
First Order ◽  
Order Language ◽  
Binary Relation Symbol ◽  
Made In

AbstractWe confirm a conjecture, about neat embeddings of cylindric algebras, made in 1969 by J. D. Monk, and a later conjecture by Maddux about relation algebras obtained from cylindric algebras. These results in algebraic logic have the following consequence for predicate logic: for every finite cardinal α ≥ 3 there is a logically valid sentence X, in a first-order language ℒ with equality and exactly one nonlogical binary relation symbol E, such that X contains only 3 variables (each of which may occur arbitrarily many times), X has a proof containing exactly α + 1 variables, but X has no proof containing only α variables. This solves a problem posed by Tarski and Givant in 1987.

scholarly journals On representable relation algebras.

1964 ◽  
Vol 11 (3) ◽  
pp. 207-210 ◽  
Author(s):  
Donald Monk
Keyword(s):  
Relation Algebras ◽  
Representable Relation

Amalgamation in relation algebras

1998 ◽  
Vol 63 (2) ◽  
pp. 479-484 ◽  
Author(s):  
Maarten Marx
Keyword(s):  
Modal Logic ◽  
Equational Theory ◽  
Weak Form ◽  
Boolean Algebras ◽  
The Other ◽  
Relation Algebras ◽  
Boolean Algebras With Operators ◽  
Other Hand ◽  
Arrow Logic ◽  
Representable Relation

We investigate amalgamation properties of relational type algebras. Besides purely algebraic interest, amalgamation in a class of algebras is important because it leads to interpolation results for the logic corresponding to that class (cf. [15]). The multi-modal logic corresponding to relational type algebras became known under the name of “arrow logic” (cf. [18, 17]), and has been studied rather extensively lately (cf. [10]). Our research was inspired by the following result of Andréka et al. [1].Let K be a class of relational type algebras such that(i) composition is associative,(ii) K is a class of boolean algebras with operators, and(iii) K contains the representable relation algebras RRA.Then the equational theory of K is undecidable.On the other hand, there are several classes of relational type algebras (e.g., NA, WA denned below) whose equational (even universal) theories are decidable (cf. [13]). Composition is not associative in these classes. Theorem 5 indicates that also with respect to amalgamation (a very weak form of) associativity forms a borderline. We first recall the relevant definitions.

The equational theory of CA3 is undecidable

1980 ◽  
Vol 45 (2) ◽  
pp. 311-316 ◽  
Author(s):  
Roger Maddux
Keyword(s):  
Equational Theory ◽  
Cylindric Algebra ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
3 Dimensional ◽  
First Order ◽  
Equational Theories ◽  
Order Language ◽  
Theoretic Proof

There is no algorithm for determining whether or not an equation is true in every 3-dimensional cylindric algebra. This theorem completes the solution to the problem of finding those values of α and β for which the equational theories of CAα and RCAβ are undecidable. (CAα and RCAβ are the classes of α-dimensional cylindric algebras and representable β-dimensional cylindric algebras. See [4] for definitions.) This problem was considered in [3]. It was known that RCA0 = CA0 and RCA1 = CA1 and that the equational theories of these classes are decidable. Tarski had shown that the equational theory of relation algebras is undecidable and, by utilizing connections between relation algebras and cylindric algebras, had also shown that the equational theories of CAα and RCAβ are undecidable whenever 4 ≤ α and 3 ≤ β. (Tarski's argument also applies to some varieties K ⊆ RCAβ with 3 ≤ β and to any variety K such that RCAα ⊆ K ⊆ CAα and 4 ≤ α.)Thus the only cases left open in 1961 were CA2, RCA2 and CA3. Shortly there-after Henkin proved, in one of Tarski's seminars at Berkeley, that the equational theory of CA2 is decidable, and Scott proved that the set of valid sentences in a first-order language with only two variables is recursive [11]. (For a more model-theoretic proof of Scott's theorem see [9].) Scott's result is equivalent to the decidability of the equational theory of RCA2, so the only case left open was CA3.

scholarly journals Undecidable properties of finite sets of equations

1976 ◽  
Vol 41 (3) ◽  
pp. 589-604 ◽  
Author(s):  
George F. McNulty
Keyword(s):  
Decision Problems ◽  
Equational Logic ◽  
Relation Algebras ◽  
Finite Sets ◽  
Operation Symbol ◽  
First Order ◽  
Finite Models ◽  
Order Language

Though equations are among the simplest sentences available in a first order language, many of the most familiar notions from algebra can be expressed by sets of equations. It is the task of this paper to expose techniques and theorems that can be used to establish that many collections of finite sets of equations characterized by common algebraic or logical properties fail to be recursive. The following theorem is typical.Theorem. In a language provided with an operation symbol of rank at least two, the collection of finite irredundant sets of equations is not recursive.Theorems of this kind are part of a pattern of research into decision problems in equational logic. This pattern finds its origins in the works of Markov [8] and Post [20] and in Tarski's development of the theory of relation algebras; see Chin [1], Chin and Tarski [2], and Tarski [23]. The papers of Mal′cev [7] and Perkins [16] are more directly connected with the present paper, which includes generalization of much of Perkins' work as well as extensions of a theorem of D. Smith [22]. V. L. Murskii [14] contains some of the results below discovered independently. Not all known results concerning undecidable properties of finite sets of equations seem to be susceptible to the methods presented here. R. McKenzie, for example, shows in [9] that for a language with an operation symbol of rank at least two, the collection of finite sets of equations with nontrivial finite models is not recursive. D. Pigozzi has extended and elaborated the techniques of this paper in [17], [18], and [19] to obtain new results concerning undecidable properties, particularly those of algebraic character.

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